Cognitive Science 301

Applied Cognitive Science and Education
Fall 2006

Mathematics, Chapter 11 (continued)

Stages of Learning of Mathematics (Smith and Rivera, 1991).  Knowing these stages is important in assessing an individual's mathematical progress.

• acquisition of skills

• proficiency (fluency after adequate practice of acquired skills)

• maintenance (need for periodic practice)

• generalization (flexible application of skills)

• adaptation (application of skills and concepts for "problem solving, reasoning, and decision making", p 417)

Assessment for specific developmental stages of learning (p 417)

• Assessment at each stage should include many, if not all, of the concepts listed for that developmental stage.

• Preschool and kindergarten

  • familiarity with number concepts

  • counting skills

  • appreciation of size and shape

  • vocabulary (more, less, equal, half)

  • sequencing

  • introduction to time and money concepts

  • introduction to addition and subtraction concepts

• Early elementary school

  • mastery of addition and subtraction skills

  • attention to mathematical signs

  • graphomotor skill at arranging numbers in columns

  • understanding place value

  • understanding regrouping

  • introduction to the concepts of multiplication in third grade

  • estimation skills

  • Need to assess:

    • conceptual understanding

    • retrieval of facts and procedures

    • application of procedures

• Late elementary and middle school

  • automatizing math facts

  • increasing skills with multiplication and long division

  • multistep problems

  • fractions, decimals, percentages

  • verbal and graphic representations/explanations of all the above

  • greater understanding of solving word problems

    • how many steps are necessary

    • what is asked for, what is not relevant

    • how does one do the calculation

  • how much does the student rely on mental arithmetic

  • rounding off numbers

  • increasing awareness and mastery of geometric concepts

  • students need to "go beyond the calculations of early elementary school and to operate on a more symbolic and abstract level" (p 421)

  • experience with calculators and computers

• Secondary school

  • algebra, trigonometry, calculus require abstract symbolic thinking

  • specialized vocabulary

  • knowledge of and retrieval of formulas

  • ability to transition between decimals, fractions, percentages

  • graphing

  • plotting equations

  • Is a solid conceptual base present?

  • Is retrieval of math facts fluent?

  • application of conceptual math to everyday problems

  • statistics and data collection

  • What would an accurate assessment contain?

Management for specific neurodevelopmental dysfunctions (pp 427-430)

• Review these (we've already discussed many of these multiple times)

• Attention deficits

• Spatial ordering deficits

• Sequential ordering weaknesses

• Memory dysfunctions

• Language disability

• Higher cognitive weaknesses

• Graphomotor dysfunction

Main ideas

1. Assessment of mathematics strengths and weaknesses must include a review of all concepts that should be acquired at each developmental stage of mathematics learning.

2. Because math proficiency builds on previously learned skills, weaknesses can develop at later stages because of inadequate acquisition of skills from previous developmental stages.

3. Proficiency is a stage of learning of mathematics which demands fluency.  Practice is also required.

4. Review how neurodevelopmental dysfunctions influence each stage of mathematics learning.

Questions

1. List the different stages of learning any math skill (p 417).  Are all math skills learned using these criteria?  Explain.

2. If an individual in college is math-phobic, but wants to improve, explain how you would go about assisting this individual in achieving success in acquisition of math skills.

3. What types of information do the different math assessments that we did in class (Math Fluency, Applied Problems, Computation) give to the person analyzing the testing results?

4. Why is math fluency stressed in elementary school?  Do you think that an individual can be strong in mathematics if their math fluency is below grade level?  Explain.